A stabilized structured Dantzig-Wolfe decomposition method

We present an algorithmic scheme, which we call the Structured Dantzig-Wolfe decomposition method, that can be used for solving large-scale structured Linear Programs (LPs). The required structure of the LPs is the same as in the original DantzigWolfe approach, that is, a polyhedron over which optimization is “easy” plus a set of “complicating” constraints. Under the hypothesis that an alternative model for the “easy” polyhedron and a simple map between the formulations are available, we propose a finite procedure which solves the original LP. The original Dantzig-Wolfe decomposition method is easily seen to be a special case of the new procedure corresponding to the standard choice for the model of the “easy” polyhedron where there is a variable for each of its extreme points and rays. Furthermore, some techniques developed for the standard DW decomposition can be easily adapted to the new approach, giving rise to Stabilized Structured Dantzig-Wolfe decomposition methods that can be more efficient than their non-stabilized brethren. We show the usefulness of our new approach with an application to the multicommodity capacitated network design problem, exploiting some recent results which characterize the polyhedral properties of the multiple choice formulation of the problem.